(i) Dynamical Stability Surveys
(ii) Capture Experiments with Growing Planetary Mass
(iii) Planetesimal-Satellite Interaction during Planetary Migration
Sun and 4 planets (Jupiter, Saturn,Uranus, Neptune). Initial conditions at the epoch J2000, correction for the barycentrum of the inner solar system. Reference plane is the invariant plane (perpendicular to the total angular momentum of 8 planets -Mercury to Neptune- at epoch J2000)
The results are shown as a `bar' code: there are
four vertical line segments at each a,i value which lengths are proportional
to the number of surviving test particles at four eccentricity values (0,
0.25, 0.5 and 0.75). The full length of 0.1 (in the scale on Y-axis) means
that all 10 integrated tps survived. When no particle survives at a,e,i
, a dot is plotted at the position of the initial orbit. Observed irregular
satellites are also shown in the figures (see the plot
of irregular satellites for more info).
These plots show several common characteristics:
Phase portrait for tps started at
a=0.355AU, e=0.5 and i=0
There exists certain asymmetry between prograde and retrograde orbits, the retrograde orbits being generally stable to larger semimajor axes that the prograde ones. It has been shown in thesis of J. Alvarellos that prograde orbits have larger heliocentric angular momentum than the retrograde ones (at the same planetocentric a). This assigns to the distant prograde satellites (a>0.45-0.48 Rh ) a value of the heliocentric Jacobi constant, which corresponds to `open' zero velocity curves: a prograde satellite with such planetocentric orbit is allowed to escape to a heliocentric orbit. Conversely, for a distant retrograde satellite (0.5<a<0.74-0.9Rh), the zero velocity curves are `closed', prohibiting escapes (at least in the basic model with the Sun on circular orbit and neglecting all other other planets). This argument shows that distant prograde satellites MAY escape but do not demonstrate WHY their orbits are chaotic and HOW the escape is achieved.
The following figure shows the orbital elements
of one escaping test particle in the numerical survey of Jupiter satellites.
This test particle started on prograde orbit with a=0.5 Rh=0.1775 AU,
Orbital elements of a distant prograde satellite affected by evection resonance.
Note the third panel where is the angle computed as the perihelion longitude of the satellite minus the mean longitude of the Sun. This angle librates during most of the seen evolution. In fact, this happens due to so-called `evection' resonance, well known from the theory of the Moon's motion (see also Touma & Wisdom (AJ 115, 1653-1663, 1998). This resonance is responsible for large variations of the eccentricity which eventually drive the satellite outside the planet's Hill sphere. The eccentricty variation and the evolution of the evection angle resemble the way how the secular resonance nu_6 affects asteroidal orbits (Farinella et al. 1994). If this analogy is as close as we believe, the evection resonance is mathematically decribed by `non-convex' resonant normal form, which permits an indefinite growth of one of the actions. This is why eccentricty grows up to 1 in the simulation. Other prograde test particles started at a=0.5-0.7 Rh behave similarly with shorter escape times at larger semimajor axis. The evection resonance does not occur for retrograde satellites because of their smaller perihelion longitude frequency (Saha and Tremaine 1993).
Mean motion resonances among Jovian satellites and the Sun
The red lines denote the area occupied by prominent first-order resonances (|kS-k| = 1). Other than first order resonances have negligible (<1e-3AU) widths: their location is shown by almost vertical sequence of points. The kS=6,k=-1 mean motion resonance which Saha and Tremaine (1993) found to effect the motion of Pasiphae is one of fine resonances near -0.15 AU (for retrograde orbits: minus the semimajor axis is shown).
This picture is highly idealized description of the real situation because the computation was done in model with Sun on the circular, co-planar orbit. If it were accounted for the eccentricity and inclination of the Sun, each mean motion resonance seen in the picture would `expand' into the multiplet structure of sub-resonances with different combinations of perihelion and nodal arguments. The multiplet structure of a single mean motion resonance would then be localized in a large range of the semimajor axis because, especially for the prograde motion, the perihelion and nodal longitudes are relatively fast angles (unlike the asteroidal case). Anyway, the mean motion resonances does not seem to determine `global' characteristics seen in our `rough' surveys.
Concluding on this point:the effect of the Kozai and evection resonances should result in flat and assymetric distribution of irregular satellites.
Number of surviving test particles vs. time in our
rough surveys for different planets:
Percentage of surviving tps in rough
The color coding: Jupiter (red), Saturn (green), Uranus (blue) and Neptune (white). Two lines for each planet are shown to distinguish between survivals on initially prograde and retrograde orbits. The retrograde orbits -being generally more stable- always correspond to the upper curve. Note that (especially for Jupiter) most test particles escape at t<10^4 yr.
To decide which is the relevant physical model to be use in the surveys, we have performed several additional runs:
|% of escapes|
|standart case (r<0.0125 AU)||(A) q<R_Jupiter||(B) q<R_Jupiter, J2 included||(C) q<R_Jupiter, Ganymede and Callisto as massive bodies|
|Prograde set||78.7 (55 far, 23.6 close)||64.2 (59.1 far, 5.1 close)||70.9 (67 far, 3.9 close)||74.8 (62.3 far, 9 close, 1.7 impact Ganymede, 1.8 impact Callisto)|
|Retrograde set||68.5 (41.2 far, 27.3 close)||55.1 (49.2 far, 5.8 close)||60.7 (55.2 far, 5.5 close)||64.3 (49 far, 7.9 close, 1.8 impact Ganymede, 5.6 impact Callisto|
The only region where the statistics of escapes substantially changes are highly inclined (i>60 deg) orbits with small semimajor axes (a~0.1-0.3 Rh). Eliminating test particles on Callisto-crossing orbits (standart case) proves to be a reasonable approximation, because both the final statistics (compare with (C)) and the stability/escape boundary are about the same as in the case with Galilean satelites physically present. It is interesting to note that the inclusion of J2 for Jupiter substantially changes the statistics (compare (A) and (B)). J2 decreases the minimum perihelion distance attained during theKozai cycle.
Osculating and mean elements of Jupiter retrogrades*
1) The Carme family has 6 members: J11 Carme (40km
diameter), S/2000 J2, J4, J6, J9, J10 (possibly also S/2000_J3)
2) The Ananke family has 3 members: J12 Ananke (30 km diameter) , S/2000 J5, J7
3) Object close to J8 Pasiphae (denoted by `P') is S/2000_J8 . `S' denotes position of J9 Sinope.
The tight groupings of satellites in the mean elements space strongly suggest their common origin.
The velocity needed to reproduce the full extent of Ananke family in the semi-major axis is 17 m/s, Carme family needs 36 m/s. For comparison, the escape velocities are ~18m/s and ~13.5m/s for Carme and Ananke, respectively (assuming 40km and 30km diameters and 1.5g/cm3 density). If collisions produced these families, they must have been `gentle' collisions, without much transfer of the linear momentum from the projectile to the target. Alternatively, families may have resulted from gas or tidal disruptions. It might be so, that the above figure also indicates that the spectral heterogenity (Brown, Michael E., AJ 119, 977-983, 2000. Luu, J., AJ 102, 1213-1225, 1991, Sykes et al., Icarus 143, 371-375, 2000) among large retrograde satellites of Jupiter is simply due to the fact that they did not originated from the same body.
*The mean inclinations of Carme, Ananke, Sinope,
and Pasiphae shown here are inclinations with respect to the local Laplacian
planes, while mean inclinations of other satellites are referred to the ecliptic.
I believe, not being sure, that the local Laplacian planes and ecliptic should
be almost identical (within ~1 degree, due to the inclination of Jupiter
orbit to ecliptics) for distant retrograde satellites, because their secular
precessions of node are primarilly driven by the Sun. Thus referring the mean
orbital elements of all satellites to the same plane should not make a big
difference in the above figure.
The following two figures (make your netscape window
large to see them side by side, if possible) show the structure of trajectories
(SIGMA_1/1 = mean longitude of a small body minus the mean longitude
of Jupiter) around the proto-Jupiter having 30 Earth masses (left) and around
Jupiter with its actual mass (right). Six panels per case are shown with
different heliocentric eccentricities (0 to 0.16 from (a) to (f)). All shown
trajectories lie in the heliocentric orbital plane of Jupiter, which is assumed
to have a circular orbit around the Sun. The red curve is where small bodies
impacts collide with Jupiter. The domain of retrograde satellites is in the
interior of this curve.
The set-up of the experiment with growing mass of the Jupiter was the following:
This experiment resulted in 10 permanent satellites! By permanent satellite we mean a case of a body which orbited Jupiter (had negative planetocentric energy) for at least past 100 yr at t=10 000 y. Visual inspection of the planetocentric orbital elements shows that all 10 permanently captured bodies were on retrograde orbits! Their orbital elements were the following:
|No.||Initial heliocentric semimajor axis (AU)||Time of the capture (yr)||Mean* planetocentric elements at t=10 000 yr||Mean* planetocentric elements when captured||Closest approach to Jupiter (Rj)|
|semimajor axis (AU)||eccentricity||inclination (deg)||perihelion dist.(AU)||semimajor axis (AU)||eccentricity||inclination (deg)|
The semimajor axis of Callisto is 0.0125 AU. The
satellites which perihelion at t=10 000 yr is smaller than 0.0125 AU
(blue color) is, according to our preliminary surveys, unstable. From
the satellites marked in red, object no. 5210 was Jupiter-grazing at
some instant of its orbital history (see last column), which would probably
have a destructive effect on its body. Objects no. 2343 and 6175
are only marginally stable at the end of simulation because of their large
eccentricites. The object which most resembles the observed Jovian retrograde
satellites is No. 5098. This figure shows its Jovicentric orbital elements
after being captured:
Planetocentric orbital elements of object No. 5098.
This body attains at t=10 000 yr a slightly smaller semimajor axis than 0.1-0.16 AU which would be typical for a real retrograde satellite, but note that at 7000<t<9000 yr the semimajor axis was in this range Consequently, if capture occured somewhat later, the final orbital elements would perfectly mimic the one of a real satellites resembling J12 Ananke.
In total, 30 temporary captures happen with duration > 50 yr. The post-capture planetocentric orbital elements of most of these bodies resemble the ones being captured permanently, but usually the inclination is <130 degrees or the semimajor axis is too large to correspond to a stable orbit. The assymetry of the stability region seen in our surveys seems to cause the preference for captures on retrograde orbits.
I have done a simple calculation: say that from 10 000 bodies between 3.8 and 7 AU only one (No.5210) resembles the orbits of retrograde satellites. The `active' interval of the initial semimajor axis where long-term captures occur is 4.5-6 AU (see Table ). Say that there is 1 good capture from 4700 test particles initially started at 2.3-6 AU. The total mass of solids in ring of 1.5 AU width at a=5 AU could have been something like 2-3.5 Earth masses. If the diferential size distribution was something like D^-3, then there were of order of 10^8-10^10 bodies with D > 4km (all 14 known retrograde satellites of Jupiter have D>4km). If we assume, according to our experience with the satellite capture due to mass growth of the planet on 10^4 years, a probability of capture ~ 10^-4 - 10^-5, we create from 1000 to 10^6 retrograde satellites. Apparently, this is too much and probably indicates that our model not good. Slower growth of the planet's mass (10^5-10^6 yr) or the presence of the gas probably decreases the capture probability. On the other hand, as Luke pointed to me, observed satelites seem to have rather shallow size-distribution (~D^-2)., so that considering D^-3 was probably not correct.One should attempt to make more serious estimates.
We conclude that our simple model with fast-growing
mass of Jupiter seems to be too efficient! Good :-)
This project has been recently started with F. Roig (visiting SwRi) and C. Beauge (staff scientist of INPE laboratory, San Jose dos Campos, Brazil). The idea is the following:
During the phase of the planetary migration, planets interacted with a large number of massive planetesimals. For example, Neptune is though to migrate as much as 7 AU. When a massive planetesimal penetrates into the Hill sphere of a planet, it certainly disturbes existing satellites. The question is how many massive planetesimals are needed to significantly change the orbit of the satellites and possibly detsabilize them. Does this set some constraint on the migration? It is hard to imagine that Neptune could have migrated by 7 AU without having lost some of its satellites due to the planetesimals gravitational sweeping. Satellites of other planets might have been also affected.
In the preliminary phase, we divide the problem in two stages:
(1) We first take the Sun and 4 outer planets on their present orbits (massive bodies) and 1000-10000 planetesimals (massles particles) initially distributed in a cold disk. We advance this system for 10-50 My and create a database of close encounters: each time a planetesimal is within a Hill sphere, its positions and velocities are registered. Later on, we assume masses for planetesimais and by the principle of action and reaction compute, from the energy and momentum balance of planetesimal's orbit, the change of planetary semimajor axis. The simulation of Hahn and Malhotra will be used as a reference. One of the nice features of this schema is that only one integration is done and different disk masses and density distribution can be considered aposteriori.This two phase procedure should be sufficiently fast to give us first hints about the process in question within few weeks.
(2) In planetocentric reference frame, advance 100-1000 satellites at different semimajor axis on their Keplerian orbits, until the first planetesimal enters within a Hill sphere. Interpolate the trajectory of the planetesimal between the points registered in the database of phase I and simulate its effect on the (massless) satellites. After planetesimal exists, advance satellites on Keplerian orbits towards the next encounter. This should be fast. The expected effect on satellites is a slow random walk of their orbital elements with possibly more violent evolution during very close planetesimal encounters.